latex debug

Author: avaamini

lab2/Part2_Debiasing.ipynb

@@ -478,32 +478,28 @@
         "In practice, how can we train a VAE? In learning the latent space, we constrain the means and standard deviations to approximately follow a unit Gaussian. Recall that these are learned parameters, and therefore must factor into the loss computation, and that the decoder portion of the VAE is using these parameters to output a reconstruction that should closely match the input image, which also must factor into the loss. What this means is that we'll have two terms in our VAE loss function:\n",
         "\n",
         "1.  **Latent loss ($L_{KL}$)**: measures how closely the learned latent variables match a unit Gaussian and is defined by the Kullback-Leibler (KL) divergence.\n",
-        "2.   **Reconstruction loss ($L_{x}{(x,\\hat{x})}$)**: measures how accurately the reconstructed outputs match the input and is given by the $L^1$ norm of the input image and its reconstructed output.  \n",
-        "\n",
-        "The equations for both of these losses are provided below:\n",
-        "\n",
-        "$$L_{KL}(\\mu, \\sigma) = \\frac{1}{2}\\sum\\limits_{j=0}^{k-1}\\small{(\\sigma_j + \\mu_j^2 - 1 - \\log{\\sigma_j})}$$\n",
-        "\n",
-        "$$L_{x}{(x,\\hat{x})} = ||x-\\hat{x}||_1$$\n",
-        "\n",
-        "Thus for the VAE loss we have: \n",
-        "\n",
-        "$$L_{VAE} = c\\cdot L_{KL} + L_{x}{(x,\\hat{x})}$$\n",
-        "\n",
-        "where $c$ is a weighting coefficient used for regularization. \n",
-        "\n",
-        "Now we're ready to define our VAE loss function:"
+        "2.   **Reconstruction loss ($L_{x}{(x,\\hat{x})}$)**: measures how accurately the reconstructed outputs match the input and is given by the $L^1$ norm of the input image and its reconstructed output."
       ]
     },
     {
       "cell_type": "markdown",
       "metadata": {
-        "id": "3UG8Ms5svZMX"
+        "id": "qWxOCPgvv1lf"
       },
       "source": [
-        "$$\r\n",
-        "1 + 2\r\n",
-        "$$"
+        "The equations for both of these losses are provided below:\r\n",
+        "\r\n",
+        "$$L_{KL}(\\mu, \\sigma) = \\frac{1}{2}\\sum\\limits_{j=0}^{k-1}\\small{(\\sigma_j + \\mu_j^2 - 1 - \\log{\\sigma_j})}$$\r\n",
+        "\r\n",
+        "$$L_{x}{(x,\\hat{x})} = ||x-\\hat{x}||_1$$\r\n",
+        "\r\n",
+        "Thus for the VAE loss we have: \r\n",
+        "\r\n",
+        "$$L_{VAE} = c\\cdot L_{KL} + L_{x}{(x,\\hat{x})}$$\r\n",
+        "\r\n",
+        "where $c$ is a weighting coefficient used for regularization. \r\n",
+        "\r\n",
+        "Now we're ready to define our VAE loss function:"
       ]
     },
     {